For centuries, scientists have attempted to identify and document analytical laws that underlie physical phenomena in nature. Despite the prevalence of computing power, finding natural laws and their corresponding equations has resisted automation. A key challenge to finding analytic relationships automatically is defining algorithmically what makes a correlation in observed data important and insightful. We have developed a technique for extracting the laws of nature from experimental data by identifying invariant and conservation equations. We demonstrate this approach by automatically searching motion-tracking data captured from various physical systems, ranging from simple harmonic oscillators to chaotic double-pendula. Without any prior knowledge about physics, kinematics or geometry, the algorithm discovered Hamiltonians, Lagrangians, and other laws of geometric and momentum conservation. The discovery rate accelerated as laws found for simpler systems were used to bootstrap explanations for more complex systems, gradually uncovering the "alphabet" used to describe those systems.